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Theorem dfon2lem1 34824
Description: Lemma for dfon2 34833. (Contributed by Scott Fenton, 28-Feb-2011.)
Assertion
Ref Expression
dfon2lem1 Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}

Proof of Theorem dfon2lem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 truni 5281 . 2 (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}Tr 𝑦 → Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)})
2 nfsbc1v 3797 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
3 nfv 1917 . . . . 5 𝑥Tr 𝑦
4 nfsbc1v 3797 . . . . 5 𝑥[𝑦 / 𝑥]𝜓
52, 3, 4nf3an 1904 . . . 4 𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓)
6 vex 3478 . . . 4 𝑦 ∈ V
7 sbceq1a 3788 . . . . 5 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
8 treq 5273 . . . . 5 (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
9 sbceq1a 3788 . . . . 5 (𝑥 = 𝑦 → (𝜓[𝑦 / 𝑥]𝜓))
107, 8, 93anbi123d 1436 . . . 4 (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓)))
115, 6, 10elabf 3665 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓))
1211simp2bi 1146 . 2 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)} → Tr 𝑦)
131, 12mprg 3067 1 Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}
Colors of variables: wff setvar class
Syntax hints:  w3a 1087  wcel 2106  {cab 2709  [wsbc 3777   cuni 4908  Tr wtr 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-v 3476  df-sbc 3778  df-in 3955  df-ss 3965  df-uni 4909  df-iun 4999  df-tr 5266
This theorem is referenced by:  dfon2lem3  34826  dfon2lem7  34830
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